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A calculation of internal kinetic energy and polarizability of compressed argon from the statistical atom model
Physica XVIII, no 11
November 1952
A CALCULATION OF I N T E R N A L K I N E T I C E N E R G Y AND P O L A R I Z A B I L I T Y OF C O M P R E S S E D ARGON FROM T H E STATISTICAL ATOM MODEL by C. A. T E N SELDAM and S. R. DE GROOT 130th publication of the Van der Waals Fund Van der Waals Laboratorium, Universiteit, Amsterdam, Nederland IHstituut voor Thcoretische Natuurkunde, tJniversiteit, Utrecht, Nederland
Synopsis From Jensell's and Gombas' m o d i f i c a t i o n of t h e s t a t i s t i c a l T h o m a s - F e r m i a t o m m o d e l , a t h e o r y for c o m p r e s s e d a t o m s is d e v e l o p e d b y changing the boundary conditions. Internal kinetic energy and polarizab i l i t y of a r g o n are c a l c u l a t e d as f u n c t i o n s of p r e s s u r e . A t 1000 a t m . a n i n t e r n a l k i n e t i c e n e r g y of a b o u t 5.2 k c a l / m o l e is f o u n d ; t h e p o l a r i z a b i l i t y is a b o u t 2 . 9 % less t h a n t h a t a t zero p r e s s u r e . T h e c h a n g e w i t h p r e s s u r e of t h e s e q u a n t i t i e s is in r o u g h a g r e e m e n t w i t h e x p e r i m e n t a l r e s u l t s a t h i g h densities.
§ 1. Introduction. Total energy, internal kinetic energy and polarizability have been calculated as functions of pressure for models of compressed atomic hydrogen 1) ~) 3) and helium i), viz., atoms enclosed in a spherical box. In these calculations rigorous wave functions were used for atomic hydrogen 3) and a very good approximation (from variation theory) for helium i). Since experimental values of internal kinetic energy 5) and polarizability e) v) of compressed argon are now available, it was thought of importance to try to find theoretical values for this atom. The statistical atom model, in particular J e n s e n's and G 0 m b as' modification 8) 9) of Gomb~s' (G, § 11) generalized Thomas-Fermi model, seemed an appropriate starting point for these calculations. (The original Thomas-Fermi model is less satisfactory, since it gives infinite values for the polarizability of free atoms). In § 2 we give an outline of the way in which the Jensen-Gomb~s method can be adapted to a statistical model of the compressed argon atom. In § 3 --910--
INTERNAL
KIN. ENERGY
~ POLARIZABILITY
OF COIvIPRESSED
ARGON
91 1
and § 4 the internal kinetic energy and the polarizability are calculated as functions of pressure. The results are compared with the experimental values.
§ 2. Statistical model [or compressed argon. For the free atom there exists in the statistical model a sphere with radius r o, outside of which the electron density ~ vanishes. In that case the pressure of the electron gas is zero at the spherical boundary at ro. As a model of compressed argon we take the statistical model with a value of r 0 smaller than that for the free atom. At the sphere with radius r o the electron gas has then a pressure Po, which is different from zero, and which is supposed to be balanced by the external pressure. In the theory of J e n s e n-G o m b ~ts for the free atom there is a condition which follows from the vanishing of Po at r o. In our case of the compressed atom this condition has to be replaced by one which pertains to a non-vanishing Po. For the free atom a certain reduced electric potential ~0(r) corresponds to zero pressure Po at radius r o. Now for the compressed atom we take a different function ~, which belongs to a different value of r o and to a corresponding Po :~ 0. In the actual calculation the different function ~o quoted above is obtained by changing the parameter k of (G 10.10) from reference '~). Then r 0 (or the corresponding reduced x 0 of G. p. 94) and ~(x0) follow from (G. 10.9). (Furthermore the electron density o, especially Oo ~ O(Xo), can be derived from (G 7.19)). Finally Po, which is equal t o - - (1/4 ~r~) (dE/dro) (E is the energy), is calculated from (G. I 1.12). This formula does not quite fit in the Jensen-Gomb~s modification, but rather in Gomb.4s' generalized Thomas-Fermi model (G, § l l). However, also Jensen uses this formula in his theory for the free atom (compare G, p. 93. We also, with Gombdts, use in (G 11.12) always the )lo calculated for the free atom ; i.e., we use the simplification of the generalized Thomas-Fermi model given in G, p. 99 ff.). TAI3LE Quantity Unit
k
.to
10-"
--
r,, tl~,
11.10 11.10 11.)5 10.99 10.30
3.895 3.895 3.878 3.857 3.790
6.0416 6.06 7 8 12
Po e2aa -4
0 3.0 1.574 3.347 1.1324
I 1
Po
~ atm. 0 1 8.70 i 457 972
I0 -s 10 -n 10 -~ 10 -~ , 3 2 8 7
. IN
kcal 'mole 0 .048 2.48 5.09 15.34
~
A~
¢1 eLt
-
2.9604 I 1 2.9596 I .9997 2.921 ] .937 2.878 i .972 2.733 I .923
912
C . A . TEN SELDAM AND S. R. DE GROOT
In table I the corresponding values of k, x 0, r 0 and P0 are given, the latter both in atomic units e2ao 4 ~ e~°m2h - 8 and in atmospheres.
§3. T h e i n l e r n a l kinetic energy. The kinetic energy of the electron is (6 3.1) Ek ---- ~-kf e 5/3 dv. With (G 7.11) and (G 7.19) this gives E k = Bf6""~5/"-x
1/2 dx ---- 2B f0' "' 95'~ dx/x,
(2)
x~here 2B is a constant (2B = 1.35541 e2aoIZT/'~N 1I:' ( N - - 1) 1;3, with Z = atomic number, N = number of electrons) which for argon has the value 7.3604 s kcal/mole. From (2) and the functions 9 of § 2, the 5
AK
2
kcal _mole
tO
/~t
ADstatistical atom model
-in-box
5
•/'~i--1~5C 2 I
0.5
-
A, experimental
f/-i
O.l C' 0
I 2
5
I0 f 2
5
I0 z 2
5
103 2
5
104aim
Fig. 1. C h a n g e of i n t e r n a l k i n e t i c e n e r g y of a r g o n b y c o m p r e s s i o n .
kinetic energy Ek can be calculated. The change AK of E~ b y compression is an important physical quantity which can be calculated from experimental compressibility data s). We obtain it b y calculating the difference of Eh and the corresponding quantity Ek/ for the free atom: AK = Ek--Ekt
= 2 B { f $ *'° (q~5/2--9~/"-) d r ' x - - f , ' . ( : / q 2 ~
I'- dVx}.
(3)
Here the index / indicates quantities of the free atom. The results of the calculation are given in table I and fig. 1. In this figure are also given experimental values for argon and, for comparison, theoretical results for compressed helium with the model of the atom enclosed in a spherical box 4). The scales for AK and P are pseudo-logarithmic., i.e., linear in log (AK + 1) and log (P + 1), respectively. Only
INTERNAL KIN. ENERGY ~ POLARIZABILITY OF COMPRESSED ARGON 913
at high pressures is the trend of the experimental and theoretical curves comparable. This is understandable since the model describes essentially repulsion at high densities but ignores attractive forces. § 4. The polarizability. We use the formula (G. 28.7) for the electric polarizability, which is valid for the Jensen-Gomb~s modification of the statistical model: a ---- K2(ro)/{2f~ ° K(r)o'I3(r) dr + (Sx,/6~e 2) K(ro)}, .
(4)
with K(r) = f~ O'/3(s) s 4 ds.
(5)
Using (G 7.11) and (G 7.19) we write (4) for argon as follows: a ---- 3.39144 10-3 Z2(Xo)/{Z(Xo) 4- 1.05882co(x0)},
(6)
with
Z(x) = f6,~ ¢p~;~(y)
(7)
ym. dy,
and with
o(x) = f0• z(y) ¢~2(y) y,~_. dy,
(8)
where a is in cubic Angstrom units. We have calculated a from (6) with the help of 9 from § 2. The results are given in table I and fig. 2. In fig. 2 the calculated quantity a/a~, where a/is the polarizability 1.005
I .OOC'
5oOc ~ O°C A, e x p e r i m e n t a l
0.995
\
0.990
A~statistical
C~885
I
0
I
I 2
i
5
'
I
I
I01 2
i
5
'
i
I
IO a 2
~
i
5
I
~
IO s 2
atom
modcl
P [
5
104arm
Fig. 2. T h e p o l a r i z a b i l i t y of the c o m p r e s s e d a r g o n a t o m . Physica XVIII
58
91 4 INTERNAL KIN. ENERGY~ POLARIZABILITY OF COMPRESSED ARGON
of the free atom, is plotted as a function of the pressure P0- The experimental values of a/a~ have been calculated from the measured dielectric constants e) by means of Kirkwood's statistical theory 7). Just as in § 3, only the change with pressure at high densities is well explained by the theory. This work is part of the research programme of the Stichting voor Fundamenteel Onderzoek der Materie. The latter foundation is financially supported by the Netherlands Organization for Pure Research (Z.W.O.). Received 13-10-52.
ERRATUM I n fig. 2, p. 913, for "0°C '' read " 2 5 ° C " ; for "150°C '' r e a d "125°C ''.
R EFE RENC [:S 1) 2) 3) 4) 5)
Miehels, A., Sommerfeld, de Groot, S. ten Seldam, Michels, A.,
d e B o e r , J . , a n d B i j l , A . , P h y s i e a 4 ( 1 9 3 7 ) 981. A. and W e l k e r , H . , A n n . Physik[5] 3 ° (1938) 56. R. and t e n S e l d a m , C. A., Physiea 12 (1946) 669. C. A. and d e G r o o t , S. R., Physica(ir~thepress). Lunbeck, R . J . and W o l k e r s , G . J . , Appl. sei. Res. A 2
(1951) 345. 6) M i e h e l s , A., t e n S e l d a m , C. A. and O v e r d i j k , S. D. J., Physiea 17 (1951) 7Sl. 7) d e B o e r , J., v a n d e r M a e s e n , F. and t e n S e l d a m , C. A., Physica • (in the press). 8) J e n s e n , H., Z. Phys. 101 (1936) 141. 9) G o m b ~,s, P., Die statistische Theorie des Atoms und ihre Anwendu~tgen; Wien, Springer, 1949. We shall refer to equations, sections, etc. of this book as G, followed by the number of the item concerned.
November 1952
A CALCULATION OF I N T E R N A L K I N E T I C E N E R G Y AND P O L A R I Z A B I L I T Y OF C O M P R E S S E D ARGON FROM T H E STATISTICAL ATOM MODEL by C. A. T E N SELDAM and S. R. DE GROOT 130th publication of the Van der Waals Fund Van der Waals Laboratorium, Universiteit, Amsterdam, Nederland IHstituut voor Thcoretische Natuurkunde, tJniversiteit, Utrecht, Nederland
Synopsis From Jensell's and Gombas' m o d i f i c a t i o n of t h e s t a t i s t i c a l T h o m a s - F e r m i a t o m m o d e l , a t h e o r y for c o m p r e s s e d a t o m s is d e v e l o p e d b y changing the boundary conditions. Internal kinetic energy and polarizab i l i t y of a r g o n are c a l c u l a t e d as f u n c t i o n s of p r e s s u r e . A t 1000 a t m . a n i n t e r n a l k i n e t i c e n e r g y of a b o u t 5.2 k c a l / m o l e is f o u n d ; t h e p o l a r i z a b i l i t y is a b o u t 2 . 9 % less t h a n t h a t a t zero p r e s s u r e . T h e c h a n g e w i t h p r e s s u r e of t h e s e q u a n t i t i e s is in r o u g h a g r e e m e n t w i t h e x p e r i m e n t a l r e s u l t s a t h i g h densities.
§ 1. Introduction. Total energy, internal kinetic energy and polarizability have been calculated as functions of pressure for models of compressed atomic hydrogen 1) ~) 3) and helium i), viz., atoms enclosed in a spherical box. In these calculations rigorous wave functions were used for atomic hydrogen 3) and a very good approximation (from variation theory) for helium i). Since experimental values of internal kinetic energy 5) and polarizability e) v) of compressed argon are now available, it was thought of importance to try to find theoretical values for this atom. The statistical atom model, in particular J e n s e n's and G 0 m b as' modification 8) 9) of Gomb~s' (G, § 11) generalized Thomas-Fermi model, seemed an appropriate starting point for these calculations. (The original Thomas-Fermi model is less satisfactory, since it gives infinite values for the polarizability of free atoms). In § 2 we give an outline of the way in which the Jensen-Gomb~s method can be adapted to a statistical model of the compressed argon atom. In § 3 --910--
INTERNAL
KIN. ENERGY
~ POLARIZABILITY
OF COIvIPRESSED
ARGON
91 1
and § 4 the internal kinetic energy and the polarizability are calculated as functions of pressure. The results are compared with the experimental values.
§ 2. Statistical model [or compressed argon. For the free atom there exists in the statistical model a sphere with radius r o, outside of which the electron density ~ vanishes. In that case the pressure of the electron gas is zero at the spherical boundary at ro. As a model of compressed argon we take the statistical model with a value of r 0 smaller than that for the free atom. At the sphere with radius r o the electron gas has then a pressure Po, which is different from zero, and which is supposed to be balanced by the external pressure. In the theory of J e n s e n-G o m b ~ts for the free atom there is a condition which follows from the vanishing of Po at r o. In our case of the compressed atom this condition has to be replaced by one which pertains to a non-vanishing Po. For the free atom a certain reduced electric potential ~0(r) corresponds to zero pressure Po at radius r o. Now for the compressed atom we take a different function ~, which belongs to a different value of r o and to a corresponding Po :~ 0. In the actual calculation the different function ~o quoted above is obtained by changing the parameter k of (G 10.10) from reference '~). Then r 0 (or the corresponding reduced x 0 of G. p. 94) and ~(x0) follow from (G. 10.9). (Furthermore the electron density o, especially Oo ~ O(Xo), can be derived from (G 7.19)). Finally Po, which is equal t o - - (1/4 ~r~) (dE/dro) (E is the energy), is calculated from (G. I 1.12). This formula does not quite fit in the Jensen-Gomb~s modification, but rather in Gomb.4s' generalized Thomas-Fermi model (G, § l l). However, also Jensen uses this formula in his theory for the free atom (compare G, p. 93. We also, with Gombdts, use in (G 11.12) always the )lo calculated for the free atom ; i.e., we use the simplification of the generalized Thomas-Fermi model given in G, p. 99 ff.). TAI3LE Quantity Unit
k
.to
10-"
--
r,, tl~,
11.10 11.10 11.)5 10.99 10.30
3.895 3.895 3.878 3.857 3.790
6.0416 6.06 7 8 12
Po e2aa -4
0 3.0 1.574 3.347 1.1324
I 1
Po
~ atm. 0 1 8.70 i 457 972
I0 -s 10 -n 10 -~ 10 -~ , 3 2 8 7
. IN
kcal 'mole 0 .048 2.48 5.09 15.34
~
A~
¢1 eLt
-
2.9604 I 1 2.9596 I .9997 2.921 ] .937 2.878 i .972 2.733 I .923
912
C . A . TEN SELDAM AND S. R. DE GROOT
In table I the corresponding values of k, x 0, r 0 and P0 are given, the latter both in atomic units e2ao 4 ~ e~°m2h - 8 and in atmospheres.
§3. T h e i n l e r n a l kinetic energy. The kinetic energy of the electron is (6 3.1) Ek ---- ~-kf e 5/3 dv. With (G 7.11) and (G 7.19) this gives E k = Bf6""~5/"-x
1/2 dx ---- 2B f0' "' 95'~ dx/x,
(2)
x~here 2B is a constant (2B = 1.35541 e2aoIZT/'~N 1I:' ( N - - 1) 1;3, with Z = atomic number, N = number of electrons) which for argon has the value 7.3604 s kcal/mole. From (2) and the functions 9 of § 2, the 5
AK
2
kcal _mole
tO
/~t
ADstatistical atom model
-in-box
5
•/'~i--1~5C 2 I
0.5
-
A, experimental
f/-i
O.l C' 0
I 2
5
I0 f 2
5
I0 z 2
5
103 2
5
104aim
Fig. 1. C h a n g e of i n t e r n a l k i n e t i c e n e r g y of a r g o n b y c o m p r e s s i o n .
kinetic energy Ek can be calculated. The change AK of E~ b y compression is an important physical quantity which can be calculated from experimental compressibility data s). We obtain it b y calculating the difference of Eh and the corresponding quantity Ek/ for the free atom: AK = Ek--Ekt
= 2 B { f $ *'° (q~5/2--9~/"-) d r ' x - - f , ' . ( : / q 2 ~
I'- dVx}.
(3)
Here the index / indicates quantities of the free atom. The results of the calculation are given in table I and fig. 1. In this figure are also given experimental values for argon and, for comparison, theoretical results for compressed helium with the model of the atom enclosed in a spherical box 4). The scales for AK and P are pseudo-logarithmic., i.e., linear in log (AK + 1) and log (P + 1), respectively. Only
INTERNAL KIN. ENERGY ~ POLARIZABILITY OF COMPRESSED ARGON 913
at high pressures is the trend of the experimental and theoretical curves comparable. This is understandable since the model describes essentially repulsion at high densities but ignores attractive forces. § 4. The polarizability. We use the formula (G. 28.7) for the electric polarizability, which is valid for the Jensen-Gomb~s modification of the statistical model: a ---- K2(ro)/{2f~ ° K(r)o'I3(r) dr + (Sx,/6~e 2) K(ro)}, .
(4)
with K(r) = f~ O'/3(s) s 4 ds.
(5)
Using (G 7.11) and (G 7.19) we write (4) for argon as follows: a ---- 3.39144 10-3 Z2(Xo)/{Z(Xo) 4- 1.05882co(x0)},
(6)
with
Z(x) = f6,~ ¢p~;~(y)
(7)
ym. dy,
and with
o(x) = f0• z(y) ¢~2(y) y,~_. dy,
(8)
where a is in cubic Angstrom units. We have calculated a from (6) with the help of 9 from § 2. The results are given in table I and fig. 2. In fig. 2 the calculated quantity a/a~, where a/is the polarizability 1.005
I .OOC'
5oOc ~ O°C A, e x p e r i m e n t a l
0.995
\
0.990
A~statistical
C~885
I
0
I
I 2
i
5
'
I
I
I01 2
i
5
'
i
I
IO a 2
~
i
5
I
~
IO s 2
atom
modcl
P [
5
104arm
Fig. 2. T h e p o l a r i z a b i l i t y of the c o m p r e s s e d a r g o n a t o m . Physica XVIII
58
91 4 INTERNAL KIN. ENERGY~ POLARIZABILITY OF COMPRESSED ARGON
of the free atom, is plotted as a function of the pressure P0- The experimental values of a/a~ have been calculated from the measured dielectric constants e) by means of Kirkwood's statistical theory 7). Just as in § 3, only the change with pressure at high densities is well explained by the theory. This work is part of the research programme of the Stichting voor Fundamenteel Onderzoek der Materie. The latter foundation is financially supported by the Netherlands Organization for Pure Research (Z.W.O.). Received 13-10-52.
ERRATUM I n fig. 2, p. 913, for "0°C '' read " 2 5 ° C " ; for "150°C '' r e a d "125°C ''.
R EFE RENC [:S 1) 2) 3) 4) 5)
Miehels, A., Sommerfeld, de Groot, S. ten Seldam, Michels, A.,
d e B o e r , J . , a n d B i j l , A . , P h y s i e a 4 ( 1 9 3 7 ) 981. A. and W e l k e r , H . , A n n . Physik[5] 3 ° (1938) 56. R. and t e n S e l d a m , C. A., Physiea 12 (1946) 669. C. A. and d e G r o o t , S. R., Physica(ir~thepress). Lunbeck, R . J . and W o l k e r s , G . J . , Appl. sei. Res. A 2
(1951) 345. 6) M i e h e l s , A., t e n S e l d a m , C. A. and O v e r d i j k , S. D. J., Physiea 17 (1951) 7Sl. 7) d e B o e r , J., v a n d e r M a e s e n , F. and t e n S e l d a m , C. A., Physica • (in the press). 8) J e n s e n , H., Z. Phys. 101 (1936) 141. 9) G o m b ~,s, P., Die statistische Theorie des Atoms und ihre Anwendu~tgen; Wien, Springer, 1949. We shall refer to equations, sections, etc. of this book as G, followed by the number of the item concerned.